On Laplace-Darboux-type sequences of generalized Weingarten surfaces (Q2738205)

Starting point of this nice and well-organized paper is the following fact: Let \(\Sigma\) be a surface in the Euclidean space \(E^3\) and \(\widetilde\Sigma\) a parallel surface in distance \(\mu\). The Gaussian curvature \(\widetilde K\) and the mean curvatue \({\widetilde M\over 2}\) of \(\widetilde \Sigma\) satisfy the condition \((\mu^2- {1\over K})\widetilde K+\mu \widetilde M+1=0\), where \(K\) denotes the Gaussian curvature of \(\Sigma\). Consequently, has \(\Sigma\) constant Gaussian curvature the parallel surfaces \(\widetilde\Sigma\) are Weingarten surfaces. Motivated by this remark the author uses the preceding relation to consider two new classes of surfaces which are parametrized by two functions \(\mu\) and \(\rho\) harmonic in a specific sense. The so-called generalized Weingarten surfaces generalize the Bianchi surfaces and the surfaces of harmonic inverse mean curvature.NEWLINENEWLINENEWLINEThe author investigates several properties of these surfaces concerning among others the classical Lelieuvre formulas, the Ernst-Weyl equation, the Loewner-Konopelchenko-Rogers representation and the Laplace-Darboux transformations. In the concluding section, a Bäcklund transformation for a particular Painlevé III equation is derived and a geometric interpretation of an associated discrete Painlevé III equation is given.